Tagged Questions

Use with the (group-theory) tag. Groups describe the symmetries of an object through their actions on the object. For example, Dihedral groups of order $2n$ acts on regular $n$-gons, $S_n$ acts on the numbers $\{1, 2, \ldots, n\}$ and the Rubik's cube group acts on Rubik's cube.

This question is inspired by questions $4.1.1$ and $4.1.2$ of Dummit and Foote. The hypothesis for the first question is formulated as: "Let $G$ act on the set $A$", and the hypothesis for the second ...

In Bourbaki, Algebra I, chapter I, §5 "Groups operating on a set" paragraph 1, Bourbaki defines the operation of a group $G$ on a set $E$ as a morphism $\alpha \in G\mapsto f_\alpha \in S(E)$ ($S(E)$ ...

Let $G$ be a cyclic group acting linearly on $X := \mathbb{A}^n$. If we assume that the quotient $Y:=X/G$ is non-singular, does it follow that $Y \simeq \mathbb{A}^n$? If so, is it necessary to assume ...

A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in ...

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup:
Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...

According to some sources on (like here or here), in the theory of combinatorial species, every molecular species has a unique factorization into atomic species. As I understand it, this is equivalent ...

First off: for my purposes, let $\sf B$ be the category of finite sets with bijections, and ${\sf B}_n$ the subcategory of sets with cardinality $n$, and define a combinatorial species to be a functor ...

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...

Let $G$ denote a finite group and recall that $G$ acts transitively (on itself) if and only if for all $x,y \in G$ there is a $g \in G$ such that $gx = y$.
I am wondering if transitive may imply that ...

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple ...

$G$ is a topological group.
Definition: A subset $S$ of $G$ is said to be syndetic in $G$ provided that $G=SK$ for some compact subset $K$ of $G$.
1.If $S$ is a syndetic subgroup in $G$, then $G/S$ ...

I'm trying to prove that $ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $; $ A \longmapsto (Ae_1, ... ,Ae_k) $ (the projection from the orthogonal group to the Stiefel manifold) is a ...

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ ...

I think I am stuck with the following statement that I read on the Encyclopedia of Mathematics website regarding Isotropy representations:
"If $G$ is a Lie group acting smoothly and transitively on ...